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nLab > Latest Changes: equivariant K-theory

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 5th 2013
    • (edited Aug 13th 2014)

    added to equivariant K-theory comments on the relation to the operator K-theory of crossed product algebras and to the ordinary K-theory of homotopy quotient spaces (Borel constructions). Also added a bunch of references.

    (Also finally added references to Green and Julg at Green-Julg theorem).

    This all deserves to be prettified further, but I have to quit now.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2018

    slightly expanded the paragraph “Relation to representation theor” (here), adding mentioning also of KO0G(*)≃Rℝ(G)

    diff, v20, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2018
    • (edited Oct 2nd 2018)

    I have added statement of the remarkable fact that equivariant KO-theory of the point subsumes the representations rings over ℝ, ℂ and ℍ:

    KOnG(*)≃{0|n=7Rℍ(G)/Rℝ(G)|n=6Rℍ(G)/Rℂ(G)|n=5Rℍ(G)/Rℝ(G)|n=40|n=3Rℂ(G)/Rℍ(G)|n=2Rℝ(G)/Rℂ(G)|n=1Rℝ(G)/Rℝ(G)|n=0

    diff, v22, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2018
    • (edited Oct 9th 2018)

    added (towards the end of this subsection) the expression for c1 of a complex representation regarded as a vector bundle over BG (from the appendix of Atiyah 61)

    diff, v24, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 9th 2018

    Is this what was meant

    c1(V)=c1(∧nV)?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2018

    Yes, that’s indeed what is meant (but we may want to add superscripts to make it seem less surprising):

    First we define c1 for line bundles/1d reps, then we define c1 on any vector bundle/rep by saying that it’s the previously defined c1 of the determinant line bundle/top exterior power.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2018
    • (edited Oct 10th 2018)

    Have expanded the respective paragraph to now read like so:

    For 1-dimensional representations V the first Chern class of ˆV is just the canonical isomorphism of 1-dimensional characters with group cohomology of G and then with ordinary cohomology of the classifying space BG

    c1(^(−)):Hom(G,U(1))≃⟶H2grp(G,ℤ)≃H2(BG,ℤ),

    while for any n-dimensional representation V the first Chern class is this isomorphism applied to the nth exterior power ∧nV of V (which is a 1-dimensional representation, namely the “determinant line bundle” of widehatV, to which the previous definition of c1 applies):

    c1(V)=c1(∧nV).

    diff, v25, current

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 10th 2018

    I think I’m struggling with the grammar. So first Chern class is an equivalence, so a map? Then “the first Chern class of X” is the image of this map applied to X? But above you’re saying this image for ˆV is an isomorphism.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2018
    • (edited Oct 10th 2018)

    Sorry for being unclear. How about this:

    For 1-dimensional representations V their first Chern class c1(ˆV)∈H2(BG,ℤ) is their image under the canonical isomorphism from 1-dimensional characters in HomGrp(G,U(1)) to the group cohomology H2grp(G,ℤ) and further to the ordinary cohomology H2(BG,ℤ) of the classifying space BG:

    c1(^(−)):HomGrp(G,U(1))≃⟶H2grp(G,ℤ)≃⟶H2(BG,ℤ).

    More generally, for n-dimensional representations V their first Chern class c1(ˆV) is the previously defined first Chern-class of the line bundle ^∧nV corresponding to the n-th exterior power ∧nV of V. The latter is a 1-dimensional representation, corresponding to the determinant line bundle det(ˆV)=^∧nV:

    c1(ˆV)=c1(det(ˆV))=c1(^∧nV).

    diff, v27, current

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 10th 2018

    Much clearer!

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2018

    appended to the previous discussion the explicit formula for c1 of an n-dimensional representation V as a polynomial in its character values (here):

    c1(V)=χ(V∧n):g↦∑k1,⋯,kn∈𝕟∑nℓ=1ℓkℓ=nn∏l=1(−1)kl+1lklkl!(χV(gl))kl

    diff, v29, current

  12. Fixing a typo in the coefficient groups of equivariant KO for n = 6; Greenlees’ reference has the correct group.

    Arun Debray

    diff, v33, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2020
    • (edited Jun 15th 2020)

    I have added a References subsection (here) on equivariant topological K-theory being represented by a naive G-spectrum.

    Currently it reads as follows:

    That G-equivariant topological K-theory is represented by a topological G-space is

    This is enhanced to a representing naive G-spectrum in

    Review includes:

    • Valentin Zakharevich, Section 2.2 of: K-Theoretic Computation of the Verlinde Ring (pdf)

    In its incarnation (under Elmendorf’s theorem) as a Spectra-valued presheaf on the G-orbit category this is discussed in

    • James Davis, Wolfgang Lück, Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory, K-Theory 15:201–252, 1998 (pdf)

    diff, v37, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2020

    added pointer to:

    diff, v41, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 4th 2020
    • (edited Oct 4th 2020)

    added pointer to

    • Wolfgang Lück, Bob Oliver, Section 1 of: Chern characters for the equivariant K-theory of proper G-CW-complexes, In: Aguadé J., Broto C., Casacuberta C. (eds.) Cohomological Methods in Homotopy Theory Progress in Mathematics, vol 196. Birkhäuser 2001 (doi:10.1007/978-3-0348-8312-2_15)

    for another construction of the representing G-space for equivariant K-theory

    diff, v42, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2020

    added pointer to

    • Yimin Yang, On the Coefficient Groups of Equivariant K-Theory, Transactions of the American Mathematical Society Vol. 347, No. 1 (Jan., 1995), pp. 77-98 (jstor:2154789)

    diff, v49, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2020

    added pointer to

    • Max Karoubi, Equivariant K-theory of real vector spaces and real vector bundles, Topology and its Applications, 122, (2002) 531-456 (arXiv:math/0509497)

    diff, v49, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2020

    added a brief remark on equivariant Bott periodicity, with pointer to section 5 in:

    Where is Atiyah’s original proof?

    diff, v49, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeNov 9th 2020

    for the proof of equivariant Bott periodicty I have added pointer to page and verse in

    diff, v51, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2020

    added statement of the equivariant K-theory of projective G-spaces (here)

    diff, v52, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2020

    added the corollary (here) on equivariant complex orientation of equivariant complex K-theory

    diff, v53, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2021

    added pointer to:

    diff, v56, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2021
    • (edited Mar 13th 2021)

    added pointer to

    which already gives a classifying G-space for G-equivariant K-theory

    diff, v60, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2021
    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeOct 3rd 2021
    • (edited Oct 3rd 2021)

    am adding these pointer:

    Will add them also to rational equivariant stable homotopy theory

    [ edit: oh, and of course this also goes to rational equivariant K-theory ]

    diff, v63, current

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2021

    added pointer to:

    diff, v64, current

    • CommentRowNumber27.
    • CommentAuthorAmartya
    • CommentTimeJan 25th 2025

    Fixed the link to Sasha Merkurjev’s notes

    diff, v70, current

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